Monotone Likelihood Ratio: Definition

The monotone likelihood ratio (MLR) represents a useful data generating process; one where there’s a clear relationship between the magnitude of observed variables and the probability distribution they are drawn from. This clear relationship makes many statistical processes possible, including identifying uniformly most powerful processes.

The MLR is defined as follows:

If the ratio of the two probability density functions, f(x) and g(x) meets the following requirement, they have the monotone likelihood ratio property:

The ratio never decreases for all X.

Phraseology

If, with respect to just a particular argument x, two functions fulfil the condition above, we would say “they have the monotone likelihood ratio property in x”. When a family of distributions all have monotonic likelihood ratios with respect to a statistic T(X), we would say that family ‘has the MLR property in T(X)’.

Families of Distributions Which Satisfy the MLR

The following families satisfy the monotone likelihood ratio property for the statistic T(X)= Σ: